In mathematics, especially algebraic topology, a simplicial diagram is a diagram indexed by the simplex category (= the category consisting of all formula_1 and the order-preserving functions).
Formally, a simplicial diagram in a category or an ∞-category "C" is a contraviant functor from the simplex category to "C". Thus, it is the same thing as a simplicial object but is typically thought of as a sequence of objects in "C" that is depicted using multiple arrows
formula_2
where formula_3 is the image of formula_4 from formula_5 in "C".
A typical example is the Čech nerve of a map formula_6; i.e., formula_7. If "F" is a presheaf with values in an ∞-category and formula_8 a Čech nerve, then formula_9 is a cosimplicial diagram and saying formula_10 is a sheaf exactly means that formula_11 is the limit of formula_9 for each formula_6 in a Grothendieck topology. See also: simplicial presheaf.
If formula_8 is a simplicial diagram, then the colimit
formula_15
is called the geometric realization of formula_8. For example, if formula_17 is an action groupoid, then the geometric realization in Grpd is the quotient groupoid formula_18 which contains more information than the set-theoretic quotient formula_19. A quotient stack is an instance of this construction (perhaps up to stackification).
The limit of a cosimplicial diagram is called the totalization of it.
Augmented simplicial diagram.
Sometimes one uses an augmented version of a simplicial diagram. Formally, an augmented simplicial diagram is a contravariant functor from the augmented simplex category formula_20 where the objects are formula_21 and the morphisms order-preserving functions.